From active to passive problem

It has to be noticed that the active nature of $\omega$ has been completely ignored in the above arguments: the prediction for the structure function given by Eq. (2.31) is identical to Eq. (2.29) obtained for the passive scalar with finite life-time. The crucial hypothesis in the derivation of Eq. (2.31) is that we have assumed that the statistics of trajectories is independent of the forcing $f_{\omega}$:

$$\langle \Omega^p e^{-p \alpha T_L(r)} \rangle_{f,T_L(r)} = ... ...mega^p \rangle_f \langle e^{-p \alpha T_L(r)} \rangle_{T_L(r)}$$ (2.35)

While for the passive scalar this is trivially true, because the forcing of the passive field has no relation with the velocity field, and cannot influence fluid trajectories, in the case of vorticity this is quite a nontrivial assumption, since it is clear that forcing does affect large-scale vorticity and thus influence velocity statistics, but it can be justified by the following argument.

The random variable $\Omega$ arises from forcing contributions along the trajectories at times $s<t-T_L(r)$, when the distance between the two fluid particles is larger than the forcing correlation length $L$, whereas the exit-time $T_L$ is clearly determined by the evolution of the strain at times $t-T_L(r)<s<t$. Since the correlation time of the strain is $\alpha^{-1}$, for

$$T_{L}(r) \gg 1/\alpha$$ (2.36)

we might expect that $\Omega$ and $T_L(r)$ be statistically independent. This condition can be translated in terms of the finite-time Lyapunov exponent as:

$$r \ll L \exp(-\gamma/\alpha)$$ (2.37)

and thus at sufficiently small scales it is reasonable to consider $\omega$ as a passive field.

We remark that, were the velocity field non-smooth, the exit-times would be independent of $r$ in the limit $r \to 0$ and the above argument would not be relevant. Therefore, the smoothness of the velocity field plays a central role in the equivalence of vorticity and passive scalar statistics.